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Observation of supersoliton resonances in the modulated annular Josephson junction
PhysicsLettersAl68(1992)319_325 North-Holland
PHYSICS LETTERS A
Observation of supersoliton resonances in the modulated annular Josephson junction I.V. Vernik,
V.A. Oboznov and A.V. Ustinov’
Institute ofSolid State Physics, Russian Academy of Sciences, Chernogolovka, Moscow District 142432, Russian Federation Received 5 June 1992; accepted for publication 6 July 1992 Communicated by J. Flouquet
We report on the experimental observation of a new resonant mode in a spatially modulated annular Josephson junction with a trapped chain of solitons. We observe voltage steps in the current—voltage characteristics of the junction. These steps appear when the spatial periods of the soliton chain and the junction modulation are incommensurate. The effect is explained in terms of the motion of defects (supersolitons) along the pinned soliton chain, which were found earlier in numerical simulations by Ustinov [Phys.Lett.A 136 (1989) 155].
1. Introduction Nonlinear wave phenomena in many solid-state systems described by the perturbed sine-Gordon equation (magnetic domains, dislocations, charge density waves, etc.) have recently drawn considerable theoretical and experimental attention. Among the many systems already discussed the closest correspondence to the pure sine-Gordon dynamics was found for solitons in long quasi-one-dimensional Josephson junctions (LJJ5) [1]. The LJJ is described
Statics and dynamics of solitons in homogeneous LJJs (f(x) = 1) have been studied in much detail so far [1]. Spatial inhomogeneities bring several new interesting features into the soliton dynamics. In particular, a spatially periodic modulation of the critical current density j~, (x) in a LJJ (a regular lattice of inhomogeneities) is usually considered theoretically in the form N
~ ö(x—ia—x~),
(2)
where ~ is the normalized strength of the inhomo-
by the perturbed sine-Gordon equation [2] ço~ ço~=f(x) sin —
ç9+ aço,
—
flco~~ + y,
(1)
where ~ t) is the space and time dependent superconducting phase difference between the electrodes of the junction. The spatial coordinate x is normalized to the Josephson penetration depth ~J, the time t is normalized to the inverse plasma frequency w5’, a and flare the dissipation coefficients. In the LJJ a solitary wave solution of eq. (1) is a magnetic flux quantum (fluxon), which moves along the junction between its superconducting electrodes. The last term in eq. (1) is the normalized bias current y which acts as a driving force for the solitons. Temporary address: Physics Laboratory I, Technical University of Denmark, DK-2800 Lyngby, Denmark.
geneities, a=A/A~is the normalized spatial period of the lattice, N is the number of inhomogeneities. The regular lattice of inhomogeneities (2) gives rise to new effects, like the soliton—plasma wave resonances [3,4] and so-called supersoliton resonances [5]. In the latter case a supersoliton is assumed to be a kind of defect (particle-like or hole-like type) moving along the pinned chain of solitons, which can be also described as a dislocation (density soliton) in the chain of solitons. Supersolitons were found numerically by Ustinov [5] for an annular LJJ with a regular lattice of inhomogeneities. Malomed [6] developed the theoretical model for supersolitons (in the case fl= 0), which is described by the equation 4f
~C
—
=
~C
sn cn dn + ~ + apk k k k
0375-9601/92/s 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
—
—
—
~,
(3) 319
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where Sn, cn and dn are standard elliptic functions, ~is the collective coordinate for the soliton chain with the average spatial period L=2kK(k) (here K(k) is the complete elliptic integral of the first kind, 0< k < 1 is the elliptic modulus), G y is the effective driving force, p is the “density of mass” in the soliton chain. This new equation (3) for ~ is different from the classical sine-Gordon equation (1) and describes the deformation of the sine-Gordon soliton chain. Solitary wave solutions ofeq. (3) have been also found analytically [6]. The existence of supersolitons has been confirmed experimentally for linear LJJs with regular inhomogeneities [7]. However, in the linear geometry the motion of solitons (and of supersolitons also) is strongly influenced by the junction boundaries. The travelling waves are reflected from the boundaries back into the LJJ, which complicates the dynamics. The complicated structure of current—voltage characteristics consisting of many resonances did not allow one to study experimentally in a clean way the dynamics of solitary waves described by eq. (3). The hole-like defects (anti-supersolitons) have not been found in this first experiment [71. In order to avoid the collisions of solitons with boundaries in a LJJ, Davidson et al. [8] have proposed and realized experimentally an annular, or ring-shaped, LJJ geometry. For an annular LJJ the boundary conditions for eq. (1) are periodic: ço(l, t)=ço(0, t) +2icn,
ço~(l,t)=c~~(0, t).
(4)
Here 1= 2~R/~~ is the normalized length (circumference) of the junction, R is the junction radius, and n is the number of trapped solitons. The annular LJJ is a topologically closed system, so the number n is conserved during the experiment. Davidson et al. have successfully trapped one soliton in the annular LJJ and studied its dynamics [8,91. Recently, a new method utilizing low temperature scanning electron microscopy (LTSEM) has been realized for trapping one by one an arbitrary number of solitons in a ring [10]. In this paper for the first time we study experimentally the supersolitons in a spatially modulated annular system. In contrast to the method proposed in ref. [10] which requires unique LTSEM facilities, we demonstrate here a new and much more acces320
31 August 1992
sible procedure for trapping solitons in annular Josephson junctions.
2. Experimental For the measurements we have used niobium—lead annular Josephson tunnel junctions which were very close to the classical “Lyngby” geometry [8] (fig. 1 a). The Nb film of 200 nm thickness served as the base electrode. The tunnel barrier, grown by plasma oxidation of Nb, was covered by a 350 nm thick top Pb electrode. The structure was formed by photolithography. Twelve annular junctions of two typical dimensions with radius R = 150 and 100 jim and with width of the tunnel barrier ring W= 10 and 20 jim, respectively, were used. The critical current density 120 was i~15 A/cm2, which corresponded to jim. The sample holder was placed in a vacuum can. The sample temperature was controlled using a car~
~N5
Cb)
I S~O
Fig. 1. (a) A sketch of the annular junction geometry used in the experiment (the SiO insulatingwindow isnot shown). (b) Schematic top view of the homogeneous annular junction. (c) Schematic top view of the junctionthe with N= 5 regularly spaced inhomogeneities which provided periodic modulation along the junction.
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bon thermometer and a compact bifilarily wound heater glued close to the sample. The measurements were performed at the temperature T= T0 = 5.0 K. An important difference of the present work from SiO insulating layer consisting oftwo concentric rings previous studies [8,10] was the use of an additional which covered the edges ofthe Nb electrode (fig. lb). The tunnel Josephson barrier was formed in the window between the two SiO rings. The thickness of the SiO layer was 150 nm. The spatially resolved investigation using LTSEM showed a high homogeneity of the tunnel barrier of these window-type annular junctions [11]. On each substrate together with homogeneous junctions we have fabricated several modulated annular LJJs. In order to provide a periodic modulation of the critical current density along the LJJ, the regular lattice ofinhomogeneities was made using SiO strips placed across the junction. The strips of width 10 jim were made together with the SiO rings and configured radially (fig. 1 c). The keystone ofthis work was the method used for trapping solitons in annular LJJs. Using photolithography we prepared on an additional glass substrate one-turn niobium coils ~‘ of the same radius as the junctions. The glass substrate was placed above the silicon substrate with the annular junctions. The Nb coils were connected in series with each other and oriented coaxially with the LJJs (fig. 2a). The distance between the two substrates during the experiment was several micrometers. Using a separate current source we were able to apply the same current ‘ext through all the coils. In order to trap the magnetic flux in annular junctions the sample was heated by the heater up to a temperature T1 7 K slightly higher than the critical temperature of lead T and then cooled slowly in the presence of a certam constant current ‘ext in the Nb coils. The magnetic field Hext produced by the current ‘ext was expected to have a radial component HR in the plane of the tunnel barrier (fig. 2b). During the cooling procedure, at T~T~some magnetic flux produced by the field H~remained trapped in the tunnel barrier area of the junction [12]. At T= 7’0 < T~’the current ‘ext was switched off and the measurements ~,
~“
The idea of usingthe current coils on a separate substrate for this experiment was suggested by V.P. Koshelets.
31 August 1992
_______________________________
____________
____________________________ Nb ~
~Nb tunnel barrier area Fig. 2. (a) Top view of the annular junction with the current coil placed above it. (b) Cross-section of the junction with the current coil.
ofthe I—Vcharacteristics were performed. Using this method we were able to get several unipolar solitons trapped in each junction. The number of solitons n (proportional to the amount of trapped magnetic flux) was dependent on ~ For reproducibility of the results the trapping procedure (heating and cooling) was controlled by a computer.
3. Results Using the technique described in the previous section we have successfully trapped solitons simultaneously in many junctions using four different substrates. The experimental results described in this section show the measurements oftwo annular junctions with radius R = 100 jim. The annular junction A was a homogeneous ring (fig. lb). The annular junction B was preparedwith a regular lattice ofN= 5 inhomogeneities (fig. 1 c). The junctions A and B were located side by side on one substrate. The critical current I,. without trapped flux was about 1.0 mA for both junctions. By a sequential cooling from the temperature the presence of the 4xt in T1 thethrough Nb coilsT~in we reproducibly obtained currentnumbers of unipolar magnetic flux quanta n equal trapped in the junctions A and B. The resonant voltage steps corresponding to dif321
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ferent soliton numbers n were observed in the I—V curves. The I—V curves were symmetrical with respect to the origin. The voltages of the steps observed in the homogeneous junction A for different 4xt were quantized as expected, according to the formula [1]
v~ =
(5)
where P 0 is the magnetic flux quantum, ~is the maximum velocity of electromagnetic waves in the junction (Swihart velocity). The voltage corresponding to the first soliton step in both junctions was about 32 j.tV. Figure 3a illustrates a typical I—Vcurve of the homogeneous junction A obtained for 1= 10 mA. 0 2
0.0 0 0
7
/
0 1 V (mV)
0 2
(b)
4 3 2 1 0
_____________
0
5 ‘ext
10
15
20
(mA)
Fig. 3. (a) Current—voltage characteristic of the homogeneous junction A with n=4 trapped solitons. (b) Dependence of the number oftrapped solitons n on the trapping current i,~,.
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According to eq. (5), in this case n = 4 solitons were trapped in the junction A. If solitons are trapped in an ideally homogeneous annular junction the non-dissipative critical current I~should be ideally equal to zero. The critical current in fig. 3a is by a factor of about 102 smaller than the flux-free critical current I~of the junction. This fact provides an important indication of a quite high technological homogeneity of our annular junctions. Previous experiments on annular LJJs without insulating window [8,10] always showed a non-zero critical current (about 5—10% of i~)when the solitons were trapped in the junction, which was probably related to the pinning of solitons by technological imperfections in the tunnel barrier. However, even in annular junctions of the present window-type geometry a certain critical current (typically 3—5% of i~)was sometimes observed after the trapping procedure. We suppose that this was happening due to the redistribution of a magnetic flux occasionally trapped in the superconducting electrodes. The most probable place for such a flux trapping can be the discontinuity region of the Nb rings where the current ~ was injected in the coils (see fig. 2a). After the trapping procedure we were able to decrease the critical current by increasing the sample temperature again close to T~and applying a small bias current through the junction. Probably, after such treatment the magnetic flux trapped in the superconducting films was redistributed, thereby decreasing the soliton pinning. The dependence of the soliton number n (determined from the voltage of the step) on the trapping current ‘ext in the annular junction A is shown in fig. 3b. Apparently, for sufficiently large trapping current I~the number of trapped solitons is approximately proportional to ‘ext~ Figure 4 shows two I—V characteristics of the inhomogeneous junction B obtained with ‘ext =3 mA (a) and ‘cxt 16 mA (b). Because the trapping conditions (temperature, ‘exi, geometry of the junction and the coil) were the same for both junctions during the experiment, we expect that the dependence ul(jext) measured for the junction A should be also the same for the junction B. From fig. 3b we can conclude that the number of solitons trapped in the junction B was n=2 and n=6 for the cases (a) and (b), respectively. In contrast to the homogeneous
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___________________________ 2
0.2~-~-)
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homogeneous window-type annular LJJs has been studied recently in an experiment [11] and a cornparison with the existing soliton chain perturbation theory [13] has been made. An unforeseen crossover in the dynamical behaviorat high velocities was tentatively interpreted in terms of soliton bunching [11,14]. In the present work our main interest was to investigate multisoliton dynamics in a modulated annular LJJ. Let us discuss the I— V curves obtained for the junction B. Our experimental results may be dis-
0 0~~
‘S
V (mV)
0.2 (b)
1
________________________
0.0 0.0
0 1
0.2
V(mV) Fig. 4. (a) Current—voltage characteristics ofthe junction with a regular lattice of N= 5 inhomogeneities with n = 2 trapped solitons. (b) Current—voltage characteristics of the same junction with n = 6 trapped solitons.
junction A (where only one main step has been observed in the I—V curve, see fig. 3a), for the modulated junction B we observe several steps in the, V curves (two steps in fig. 4a and three steps in fig. 4b). The physical mechanism related to the additional steps observed in figs. 4a, 4b is discussed in the following section.
4. Discussion For a homogeneous annular LJJ the shape of the steps in the I— V curves provides information about the soliton dynamics. The multisoliton behavior in
cussed in the framework of the approach suggested in ref. [5]. For the simplest case, let us start from the completely commensurate case of equal numbers of solitons and inhomogeneities in the junction: N= n =5 (fig. 5a). Each inhomogeneity represents a region in the LJJ where the critical current density is suppressed, which corresponds to e <0 in eq. (2). Such a region (sometimes called a microresistor) attracts a soliton [15]. Because the soliton chain and the lattice of inhomogeneities are commensurate (a=L), at sufficiently low bias current all solitons are located on inhomogeneities and the chain is strongly pinned. The density of solitons is constant along the chain. If we add one soliton into the junction (n=6, which corresponds to fig. 4b), the density of solitons becomes spatially dependent [5]. It is energetically advantageous to place the sixth soliton as a defect on the already pinned regular lattice of five other solitons (fig. Sb). Under the influence ofa bias current this defect (supersoliton) can move along the junction, while all other solitons remain pinned. In the I—V curve this mechanism leads to the step at the voltage
V= V
1. In fig. 4b this gives the cxplanation for the large first step at V~32 j.tV. In a more general case, a commensurate arrangement of solitons and inhomogeneities takes place
when qa=pL, where p and q are integers, a is the normalized spatial period of inhomogeneities, L is the normalized period of the soliton chain. For a finite system the numbers p and q should be not too large, so that the commensurability period pL remains smaller than the size of the system 1. Another restriction, to have only small p and q, comes from the finite size of each inhomogeneity in real experiments. It has been shown [16] that for a very dense soliton chain with L e/1.1 even in the commensur‘~
323
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31 August 1992
posite, while the average voltage has the same sign. Coming back to fig. 4b we find that another small step in the I—V curve at the voltage V= V~ 128 j.tV can be explained as corresponding to ns = —4 (four anti-supersolitons) excited from another commensot don
surate state with p=2 and q= 1 (fig. Sc). The last step at V= V6 190 j.tV is the usual flux-flow of n = 6
—
solitons in the junction. By chance, exactly the same
(b) X
—tr—r-—g—g——g
anti- sot iton
0
—~
—
0
—
0
0
0
0
Fig. 5. Schematic presentation of different regimes for the solitons moving in a long Josephson junction with N= 5 regularly spaced inhomogeneities: (a) commensurate state with n = 5; (b) incommensurate state with n = 6 corresponding to ns = 1 (one supersoliton); (c) incommensurate state with n = 6 corresponding to ~s —4 (four anti-supersolitons); (d) incommensurate state with n=2 corresponding to ~s= —3 (three anti-supersolitons). Note that in the experiment the boundary conditions are periodic.
ate case the pinning of the soliton chain can disappear. In the incommensurate state the number of supersolitons ns may be calculated using the simple formula [51 t1s =n—Np/q, (6) where ns> 0 for supersolitons (particle-like defects) and ns <0 for anti-supersolitons (hole-like defects). In the topologically closed system (i.e. in annular geometry) only the integer values of ns are permitted. Like conventional solitons and anti-solitons in LJJs, supersolitons and anti-supersolitons have opposite polarities. The directions of their motion are also op324
configuration with n = 6 and N 5 has been already simulated numerically [5]. The present experiment is in very good agreement with those numerical results, which also showed the steps in the I—V curve at the voltages V1, V4, and V6. According to the measured dependence n (Iext) (fig. 3b) fig. 4a corresponds to n = 2 solitons trapped in the junction B. Thus, the step at V= V2 64 jiV corresponds to two solitons moving in the junction. Following the model described above, another step in the I—V curve at the voltage V= V3 96 j.tV can be explained as corresponding to ns = —3 (three antisupersolitons) excited from the main commensurate state with p = q = 1. We see that this I—V curve is also well explained by the supersoliton model. Note that here we have I ns > n. Physically, the way to obtain such a state in the annular junction with n=2 solitons is to create n~,= 3 soliton—anti-soliton pairs [8,91 and to pin n + n~= 5 solitons in a commensurate way on N= 5 inhomogeneities. With the three remaining anti-solitons we have fl~= n~,= 3 (fig. Sd). We see that the supersoliton approach gives a selfconsistent explanation for all the additional steps observed in the I—V characteristics of the spatially —
modulated annular junction B. Both the supersoliton-type and the anti-supersoliton-type steps are observed in the experiment. Here we gave only a qualitative comparison ofour experimental data with the theoretical model [5]. Taking the more generalized approach developed by Malomed [6] it would be interesting to compare quantitatively the detailed shape of the supersoliton steps with the perturbation theory following from the elliptic sine-Gordon equation (3). However, eq. (3) was obtained in the limit when the variation of the soliton density occurs on a spatial scale much larger than the period of the soliton chain L. In this sense the relatively short normalized length 2icR /2 S of the inhomogeneous junctions studied in the present work brings the basic restriction for application of .j
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the theoretical formulas obtained in the infinite spatial limit. We also want to point out the analogy between supersoliton behavior in one-dimensional modulated LJJ discussed here with recent studies of the frustrated magnetic flux states in two-dimensional j0 sephson junction arrays. The ri-induced “fractional giant Shapiro steps” found in SNS arrays by Benz et al. [171 represent another example of the same nature as the self-induced supersoliton steps found in our study. In the experiments with two- dimensional arrays the ri-field stimulates the motion of a twodimensional magnetic flux superlattice, which exists when the amount of flux per unit cell (frustration parameter]) is not an integer. The magnetic flux dynamics in our experimental system differs from that of an SNS array by a much lower dissipation exhibited by fluxons. We see the motion of a fluxon superlattice without any external stimulation as self-induced resonances on the I—V characteristics.
Acknowledgement We are grateful to T. Doderer, V.P. Koshelets, B.A. Malorned and N.F. Pedersen for stimulating discussions. We also thank V. Kaplunenko for the layout design using his “Z”-topological compiler program and A. Goncharov for the manufacturing of the photomasks. A.V.U. gratefully acknowledges partial sup-
31 August 1992
port by the Alexander von Humboldt Stiftung. References [1] N.F. Pedersen, in: Josephson effect
—
achievements and
trends, ed. A. Barone (World Scientific, Singapore, 1986). [2] D.W. McLaughlin and AC. Scott, Phys. Rev. A 18 (1978) 1652. [3] A.A. Golubov, I.L. Serpuchenko and A.V. Ustinov, Phys. Lett. A 130 (1988) 107. [4] B.A. 254. [5] A.V. [6] B.A. [7] V.A. 481.
Malomed
and A.V.
Ustinov, Phys. Rev. B 41(1990)
Ustinov, Phys. Lett A 136 (1989) 155. Malomed, Phys. Rev. B 41(1990)2616. Oboznov and A.V. Ustinov, Phys. Lett. A 139 (1989)
[8] A. Davidson, B. Dueholm, B. Kryger and N.F. Pedersen, Phys. Rev. Lett. 55 (1985) 2059. [9] A. Davidson, B. Dueholm and N.F. Pedersen, J. Appl. Phys. 60(1986)1447. [10] A.V. Ustinov, T. Doderer, B. Mayer, R.P. Huebener and V.A. Oboznov, Europhys. Lett., to be published. [11] A.V. Ustinov, T. Doderer, R.P. Huebener, N.F. Pedersen, B. Mayer and V.A. Oboznov, unpublished. [12] A.V. Ustinov, T. Doderer, B. Mayer, R.P. Huebener, IV. Vernik and V.A. Oboznov, in: SQUID’9 I — Superconducting devices and their applications, eds. H. Koch and H. Lubbig (Springer, Berlin, 1992). [13] P.M. Marcus and Y. Imry, Solid State Commun. 33 (1980) 345. [14] B.A. Malomed, unpublished. [15] Yu.S. Gal’pern and A.T. Filippov, Solid State Commun. 48 (1983) 665; Soy. Phys. JETP 86 (1984) 1527. [16] R. Fehrenbacher, Rev.B45 (1992) V.B. 5450.Geshkenbein and G. Blatter, Phys. [17] S.P. Benz, MS. Rzchowski, M. Tinkham and C.J. Lobb, Phys. Rev. Lett. 64 (1990) 693.
325
PHYSICS LETTERS A
Observation of supersoliton resonances in the modulated annular Josephson junction I.V. Vernik,
V.A. Oboznov and A.V. Ustinov’
Institute ofSolid State Physics, Russian Academy of Sciences, Chernogolovka, Moscow District 142432, Russian Federation Received 5 June 1992; accepted for publication 6 July 1992 Communicated by J. Flouquet
We report on the experimental observation of a new resonant mode in a spatially modulated annular Josephson junction with a trapped chain of solitons. We observe voltage steps in the current—voltage characteristics of the junction. These steps appear when the spatial periods of the soliton chain and the junction modulation are incommensurate. The effect is explained in terms of the motion of defects (supersolitons) along the pinned soliton chain, which were found earlier in numerical simulations by Ustinov [Phys.Lett.A 136 (1989) 155].
1. Introduction Nonlinear wave phenomena in many solid-state systems described by the perturbed sine-Gordon equation (magnetic domains, dislocations, charge density waves, etc.) have recently drawn considerable theoretical and experimental attention. Among the many systems already discussed the closest correspondence to the pure sine-Gordon dynamics was found for solitons in long quasi-one-dimensional Josephson junctions (LJJ5) [1]. The LJJ is described
Statics and dynamics of solitons in homogeneous LJJs (f(x) = 1) have been studied in much detail so far [1]. Spatial inhomogeneities bring several new interesting features into the soliton dynamics. In particular, a spatially periodic modulation of the critical current density j~, (x) in a LJJ (a regular lattice of inhomogeneities) is usually considered theoretically in the form N
~ ö(x—ia—x~),
(2)
where ~ is the normalized strength of the inhomo-
by the perturbed sine-Gordon equation [2] ço~ ço~=f(x) sin —
ç9+ aço,
—
flco~~ + y,
(1)
where ~ t) is the space and time dependent superconducting phase difference between the electrodes of the junction. The spatial coordinate x is normalized to the Josephson penetration depth ~J, the time t is normalized to the inverse plasma frequency w5’, a and flare the dissipation coefficients. In the LJJ a solitary wave solution of eq. (1) is a magnetic flux quantum (fluxon), which moves along the junction between its superconducting electrodes. The last term in eq. (1) is the normalized bias current y which acts as a driving force for the solitons. Temporary address: Physics Laboratory I, Technical University of Denmark, DK-2800 Lyngby, Denmark.
geneities, a=A/A~is the normalized spatial period of the lattice, N is the number of inhomogeneities. The regular lattice of inhomogeneities (2) gives rise to new effects, like the soliton—plasma wave resonances [3,4] and so-called supersoliton resonances [5]. In the latter case a supersoliton is assumed to be a kind of defect (particle-like or hole-like type) moving along the pinned chain of solitons, which can be also described as a dislocation (density soliton) in the chain of solitons. Supersolitons were found numerically by Ustinov [5] for an annular LJJ with a regular lattice of inhomogeneities. Malomed [6] developed the theoretical model for supersolitons (in the case fl= 0), which is described by the equation 4f
~C
—
=
~C
sn cn dn + ~ + apk k k k
0375-9601/92/s 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
—
—
—
~,
(3) 319
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where Sn, cn and dn are standard elliptic functions, ~is the collective coordinate for the soliton chain with the average spatial period L=2kK(k) (here K(k) is the complete elliptic integral of the first kind, 0< k < 1 is the elliptic modulus), G y is the effective driving force, p is the “density of mass” in the soliton chain. This new equation (3) for ~ is different from the classical sine-Gordon equation (1) and describes the deformation of the sine-Gordon soliton chain. Solitary wave solutions ofeq. (3) have been also found analytically [6]. The existence of supersolitons has been confirmed experimentally for linear LJJs with regular inhomogeneities [7]. However, in the linear geometry the motion of solitons (and of supersolitons also) is strongly influenced by the junction boundaries. The travelling waves are reflected from the boundaries back into the LJJ, which complicates the dynamics. The complicated structure of current—voltage characteristics consisting of many resonances did not allow one to study experimentally in a clean way the dynamics of solitary waves described by eq. (3). The hole-like defects (anti-supersolitons) have not been found in this first experiment [71. In order to avoid the collisions of solitons with boundaries in a LJJ, Davidson et al. [8] have proposed and realized experimentally an annular, or ring-shaped, LJJ geometry. For an annular LJJ the boundary conditions for eq. (1) are periodic: ço(l, t)=ço(0, t) +2icn,
ço~(l,t)=c~~(0, t).
(4)
Here 1= 2~R/~~ is the normalized length (circumference) of the junction, R is the junction radius, and n is the number of trapped solitons. The annular LJJ is a topologically closed system, so the number n is conserved during the experiment. Davidson et al. have successfully trapped one soliton in the annular LJJ and studied its dynamics [8,91. Recently, a new method utilizing low temperature scanning electron microscopy (LTSEM) has been realized for trapping one by one an arbitrary number of solitons in a ring [10]. In this paper for the first time we study experimentally the supersolitons in a spatially modulated annular system. In contrast to the method proposed in ref. [10] which requires unique LTSEM facilities, we demonstrate here a new and much more acces320
31 August 1992
sible procedure for trapping solitons in annular Josephson junctions.
2. Experimental For the measurements we have used niobium—lead annular Josephson tunnel junctions which were very close to the classical “Lyngby” geometry [8] (fig. 1 a). The Nb film of 200 nm thickness served as the base electrode. The tunnel barrier, grown by plasma oxidation of Nb, was covered by a 350 nm thick top Pb electrode. The structure was formed by photolithography. Twelve annular junctions of two typical dimensions with radius R = 150 and 100 jim and with width of the tunnel barrier ring W= 10 and 20 jim, respectively, were used. The critical current density 120 was i~15 A/cm2, which corresponded to jim. The sample holder was placed in a vacuum can. The sample temperature was controlled using a car~
~N5
Cb)
I S~O
Fig. 1. (a) A sketch of the annular junction geometry used in the experiment (the SiO insulatingwindow isnot shown). (b) Schematic top view of the homogeneous annular junction. (c) Schematic top view of the junctionthe with N= 5 regularly spaced inhomogeneities which provided periodic modulation along the junction.
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bon thermometer and a compact bifilarily wound heater glued close to the sample. The measurements were performed at the temperature T= T0 = 5.0 K. An important difference of the present work from SiO insulating layer consisting oftwo concentric rings previous studies [8,10] was the use of an additional which covered the edges ofthe Nb electrode (fig. lb). The tunnel Josephson barrier was formed in the window between the two SiO rings. The thickness of the SiO layer was 150 nm. The spatially resolved investigation using LTSEM showed a high homogeneity of the tunnel barrier of these window-type annular junctions [11]. On each substrate together with homogeneous junctions we have fabricated several modulated annular LJJs. In order to provide a periodic modulation of the critical current density along the LJJ, the regular lattice ofinhomogeneities was made using SiO strips placed across the junction. The strips of width 10 jim were made together with the SiO rings and configured radially (fig. 1 c). The keystone ofthis work was the method used for trapping solitons in annular LJJs. Using photolithography we prepared on an additional glass substrate one-turn niobium coils ~‘ of the same radius as the junctions. The glass substrate was placed above the silicon substrate with the annular junctions. The Nb coils were connected in series with each other and oriented coaxially with the LJJs (fig. 2a). The distance between the two substrates during the experiment was several micrometers. Using a separate current source we were able to apply the same current ‘ext through all the coils. In order to trap the magnetic flux in annular junctions the sample was heated by the heater up to a temperature T1 7 K slightly higher than the critical temperature of lead T and then cooled slowly in the presence of a certam constant current ‘ext in the Nb coils. The magnetic field Hext produced by the current ‘ext was expected to have a radial component HR in the plane of the tunnel barrier (fig. 2b). During the cooling procedure, at T~T~some magnetic flux produced by the field H~remained trapped in the tunnel barrier area of the junction [12]. At T= 7’0 < T~’the current ‘ext was switched off and the measurements ~,
~“
The idea of usingthe current coils on a separate substrate for this experiment was suggested by V.P. Koshelets.
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_______________________________
____________
____________________________ Nb ~
~Nb tunnel barrier area Fig. 2. (a) Top view of the annular junction with the current coil placed above it. (b) Cross-section of the junction with the current coil.
ofthe I—Vcharacteristics were performed. Using this method we were able to get several unipolar solitons trapped in each junction. The number of solitons n (proportional to the amount of trapped magnetic flux) was dependent on ~ For reproducibility of the results the trapping procedure (heating and cooling) was controlled by a computer.
3. Results Using the technique described in the previous section we have successfully trapped solitons simultaneously in many junctions using four different substrates. The experimental results described in this section show the measurements oftwo annular junctions with radius R = 100 jim. The annular junction A was a homogeneous ring (fig. lb). The annular junction B was preparedwith a regular lattice ofN= 5 inhomogeneities (fig. 1 c). The junctions A and B were located side by side on one substrate. The critical current I,. without trapped flux was about 1.0 mA for both junctions. By a sequential cooling from the temperature the presence of the 4xt in T1 thethrough Nb coilsT~in we reproducibly obtained currentnumbers of unipolar magnetic flux quanta n equal trapped in the junctions A and B. The resonant voltage steps corresponding to dif321
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ferent soliton numbers n were observed in the I—V curves. The I—V curves were symmetrical with respect to the origin. The voltages of the steps observed in the homogeneous junction A for different 4xt were quantized as expected, according to the formula [1]
v~ =
(5)
where P 0 is the magnetic flux quantum, ~is the maximum velocity of electromagnetic waves in the junction (Swihart velocity). The voltage corresponding to the first soliton step in both junctions was about 32 j.tV. Figure 3a illustrates a typical I—Vcurve of the homogeneous junction A obtained for 1= 10 mA. 0 2
0.0 0 0
7
/
0 1 V (mV)
0 2
(b)
4 3 2 1 0
_____________
0
5 ‘ext
10
15
20
(mA)
Fig. 3. (a) Current—voltage characteristic of the homogeneous junction A with n=4 trapped solitons. (b) Dependence of the number oftrapped solitons n on the trapping current i,~,.
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According to eq. (5), in this case n = 4 solitons were trapped in the junction A. If solitons are trapped in an ideally homogeneous annular junction the non-dissipative critical current I~should be ideally equal to zero. The critical current in fig. 3a is by a factor of about 102 smaller than the flux-free critical current I~of the junction. This fact provides an important indication of a quite high technological homogeneity of our annular junctions. Previous experiments on annular LJJs without insulating window [8,10] always showed a non-zero critical current (about 5—10% of i~)when the solitons were trapped in the junction, which was probably related to the pinning of solitons by technological imperfections in the tunnel barrier. However, even in annular junctions of the present window-type geometry a certain critical current (typically 3—5% of i~)was sometimes observed after the trapping procedure. We suppose that this was happening due to the redistribution of a magnetic flux occasionally trapped in the superconducting electrodes. The most probable place for such a flux trapping can be the discontinuity region of the Nb rings where the current ~ was injected in the coils (see fig. 2a). After the trapping procedure we were able to decrease the critical current by increasing the sample temperature again close to T~and applying a small bias current through the junction. Probably, after such treatment the magnetic flux trapped in the superconducting films was redistributed, thereby decreasing the soliton pinning. The dependence of the soliton number n (determined from the voltage of the step) on the trapping current ‘ext in the annular junction A is shown in fig. 3b. Apparently, for sufficiently large trapping current I~the number of trapped solitons is approximately proportional to ‘ext~ Figure 4 shows two I—V characteristics of the inhomogeneous junction B obtained with ‘ext =3 mA (a) and ‘cxt 16 mA (b). Because the trapping conditions (temperature, ‘exi, geometry of the junction and the coil) were the same for both junctions during the experiment, we expect that the dependence ul(jext) measured for the junction A should be also the same for the junction B. From fig. 3b we can conclude that the number of solitons trapped in the junction B was n=2 and n=6 for the cases (a) and (b), respectively. In contrast to the homogeneous
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0.2~-~-)
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homogeneous window-type annular LJJs has been studied recently in an experiment [11] and a cornparison with the existing soliton chain perturbation theory [13] has been made. An unforeseen crossover in the dynamical behaviorat high velocities was tentatively interpreted in terms of soliton bunching [11,14]. In the present work our main interest was to investigate multisoliton dynamics in a modulated annular LJJ. Let us discuss the I— V curves obtained for the junction B. Our experimental results may be dis-
0 0~~
‘S
V (mV)
0.2 (b)
1
________________________
0.0 0.0
0 1
0.2
V(mV) Fig. 4. (a) Current—voltage characteristics ofthe junction with a regular lattice of N= 5 inhomogeneities with n = 2 trapped solitons. (b) Current—voltage characteristics of the same junction with n = 6 trapped solitons.
junction A (where only one main step has been observed in the I—V curve, see fig. 3a), for the modulated junction B we observe several steps in the, V curves (two steps in fig. 4a and three steps in fig. 4b). The physical mechanism related to the additional steps observed in figs. 4a, 4b is discussed in the following section.
4. Discussion For a homogeneous annular LJJ the shape of the steps in the I— V curves provides information about the soliton dynamics. The multisoliton behavior in
cussed in the framework of the approach suggested in ref. [5]. For the simplest case, let us start from the completely commensurate case of equal numbers of solitons and inhomogeneities in the junction: N= n =5 (fig. 5a). Each inhomogeneity represents a region in the LJJ where the critical current density is suppressed, which corresponds to e <0 in eq. (2). Such a region (sometimes called a microresistor) attracts a soliton [15]. Because the soliton chain and the lattice of inhomogeneities are commensurate (a=L), at sufficiently low bias current all solitons are located on inhomogeneities and the chain is strongly pinned. The density of solitons is constant along the chain. If we add one soliton into the junction (n=6, which corresponds to fig. 4b), the density of solitons becomes spatially dependent [5]. It is energetically advantageous to place the sixth soliton as a defect on the already pinned regular lattice of five other solitons (fig. Sb). Under the influence ofa bias current this defect (supersoliton) can move along the junction, while all other solitons remain pinned. In the I—V curve this mechanism leads to the step at the voltage
V= V
1. In fig. 4b this gives the cxplanation for the large first step at V~32 j.tV. In a more general case, a commensurate arrangement of solitons and inhomogeneities takes place
when qa=pL, where p and q are integers, a is the normalized spatial period of inhomogeneities, L is the normalized period of the soliton chain. For a finite system the numbers p and q should be not too large, so that the commensurability period pL remains smaller than the size of the system 1. Another restriction, to have only small p and q, comes from the finite size of each inhomogeneity in real experiments. It has been shown [16] that for a very dense soliton chain with L e/1.1 even in the commensur‘~
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posite, while the average voltage has the same sign. Coming back to fig. 4b we find that another small step in the I—V curve at the voltage V= V~ 128 j.tV can be explained as corresponding to ns = —4 (four anti-supersolitons) excited from another commensot don
surate state with p=2 and q= 1 (fig. Sc). The last step at V= V6 190 j.tV is the usual flux-flow of n = 6
—
solitons in the junction. By chance, exactly the same
(b) X
—tr—r-—g—g——g
anti- sot iton
0
—~
—
0
—
0
0
0
0
Fig. 5. Schematic presentation of different regimes for the solitons moving in a long Josephson junction with N= 5 regularly spaced inhomogeneities: (a) commensurate state with n = 5; (b) incommensurate state with n = 6 corresponding to ns = 1 (one supersoliton); (c) incommensurate state with n = 6 corresponding to ~s —4 (four anti-supersolitons); (d) incommensurate state with n=2 corresponding to ~s= —3 (three anti-supersolitons). Note that in the experiment the boundary conditions are periodic.
ate case the pinning of the soliton chain can disappear. In the incommensurate state the number of supersolitons ns may be calculated using the simple formula [51 t1s =n—Np/q, (6) where ns> 0 for supersolitons (particle-like defects) and ns <0 for anti-supersolitons (hole-like defects). In the topologically closed system (i.e. in annular geometry) only the integer values of ns are permitted. Like conventional solitons and anti-solitons in LJJs, supersolitons and anti-supersolitons have opposite polarities. The directions of their motion are also op324
configuration with n = 6 and N 5 has been already simulated numerically [5]. The present experiment is in very good agreement with those numerical results, which also showed the steps in the I—V curve at the voltages V1, V4, and V6. According to the measured dependence n (Iext) (fig. 3b) fig. 4a corresponds to n = 2 solitons trapped in the junction B. Thus, the step at V= V2 64 jiV corresponds to two solitons moving in the junction. Following the model described above, another step in the I—V curve at the voltage V= V3 96 j.tV can be explained as corresponding to ns = —3 (three antisupersolitons) excited from the main commensurate state with p = q = 1. We see that this I—V curve is also well explained by the supersoliton model. Note that here we have I ns > n. Physically, the way to obtain such a state in the annular junction with n=2 solitons is to create n~,= 3 soliton—anti-soliton pairs [8,91 and to pin n + n~= 5 solitons in a commensurate way on N= 5 inhomogeneities. With the three remaining anti-solitons we have fl~= n~,= 3 (fig. Sd). We see that the supersoliton approach gives a selfconsistent explanation for all the additional steps observed in the I—V characteristics of the spatially —
modulated annular junction B. Both the supersoliton-type and the anti-supersoliton-type steps are observed in the experiment. Here we gave only a qualitative comparison ofour experimental data with the theoretical model [5]. Taking the more generalized approach developed by Malomed [6] it would be interesting to compare quantitatively the detailed shape of the supersoliton steps with the perturbation theory following from the elliptic sine-Gordon equation (3). However, eq. (3) was obtained in the limit when the variation of the soliton density occurs on a spatial scale much larger than the period of the soliton chain L. In this sense the relatively short normalized length 2icR /2 S of the inhomogeneous junctions studied in the present work brings the basic restriction for application of .j
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the theoretical formulas obtained in the infinite spatial limit. We also want to point out the analogy between supersoliton behavior in one-dimensional modulated LJJ discussed here with recent studies of the frustrated magnetic flux states in two-dimensional j0 sephson junction arrays. The ri-induced “fractional giant Shapiro steps” found in SNS arrays by Benz et al. [171 represent another example of the same nature as the self-induced supersoliton steps found in our study. In the experiments with two- dimensional arrays the ri-field stimulates the motion of a twodimensional magnetic flux superlattice, which exists when the amount of flux per unit cell (frustration parameter]) is not an integer. The magnetic flux dynamics in our experimental system differs from that of an SNS array by a much lower dissipation exhibited by fluxons. We see the motion of a fluxon superlattice without any external stimulation as self-induced resonances on the I—V characteristics.
Acknowledgement We are grateful to T. Doderer, V.P. Koshelets, B.A. Malorned and N.F. Pedersen for stimulating discussions. We also thank V. Kaplunenko for the layout design using his “Z”-topological compiler program and A. Goncharov for the manufacturing of the photomasks. A.V.U. gratefully acknowledges partial sup-
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port by the Alexander von Humboldt Stiftung. References [1] N.F. Pedersen, in: Josephson effect
—
achievements and
trends, ed. A. Barone (World Scientific, Singapore, 1986). [2] D.W. McLaughlin and AC. Scott, Phys. Rev. A 18 (1978) 1652. [3] A.A. Golubov, I.L. Serpuchenko and A.V. Ustinov, Phys. Lett. A 130 (1988) 107. [4] B.A. 254. [5] A.V. [6] B.A. [7] V.A. 481.
Malomed
and A.V.
Ustinov, Phys. Rev. B 41(1990)
Ustinov, Phys. Lett A 136 (1989) 155. Malomed, Phys. Rev. B 41(1990)2616. Oboznov and A.V. Ustinov, Phys. Lett. A 139 (1989)
[8] A. Davidson, B. Dueholm, B. Kryger and N.F. Pedersen, Phys. Rev. Lett. 55 (1985) 2059. [9] A. Davidson, B. Dueholm and N.F. Pedersen, J. Appl. Phys. 60(1986)1447. [10] A.V. Ustinov, T. Doderer, B. Mayer, R.P. Huebener and V.A. Oboznov, Europhys. Lett., to be published. [11] A.V. Ustinov, T. Doderer, R.P. Huebener, N.F. Pedersen, B. Mayer and V.A. Oboznov, unpublished. [12] A.V. Ustinov, T. Doderer, B. Mayer, R.P. Huebener, IV. Vernik and V.A. Oboznov, in: SQUID’9 I — Superconducting devices and their applications, eds. H. Koch and H. Lubbig (Springer, Berlin, 1992). [13] P.M. Marcus and Y. Imry, Solid State Commun. 33 (1980) 345. [14] B.A. Malomed, unpublished. [15] Yu.S. Gal’pern and A.T. Filippov, Solid State Commun. 48 (1983) 665; Soy. Phys. JETP 86 (1984) 1527. [16] R. Fehrenbacher, Rev.B45 (1992) V.B. 5450.Geshkenbein and G. Blatter, Phys. [17] S.P. Benz, MS. Rzchowski, M. Tinkham and C.J. Lobb, Phys. Rev. Lett. 64 (1990) 693.
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